A225356 Triangle T(n, k) = T(n, k-1) + (-1)^k*A060187(n+2,k+1) if k <= floor(n/2), otherwise T(n, n-k), with T(n, 0) = T(n, n) = 1, read by rows.

1, 1, 1, 1, -22, 1, 1, -75, -75, 1, 1, -236, 1446, -236, 1, 1, -721, 9822, 9822, -721, 1, 1, -2178, 58479, -201244, 58479, -2178, 1, 1, -6551, 325061, -2160227, -2160227, 325061, -6551, 1, 1, -19672, 1736668, -19971304, 49441990, -19971304, 1736668, -19672, 1

Offset

0,5

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = T(n, k-1) + (-1)^k*A060187(n+2,k+1) if k <= floor(n/2), otherwise T(n, n-k), with T(n, 0) = T(n, n) = 1.

Example

The triangle begins:
  1;
  1,      1;
  1,    -22,       1;
  1,    -75,     -75,         1;
  1,   -236,    1446,      -236,        1;
  1,   -721,    9822,      9822,     -721,         1;
  1,  -2178,   58479,   -201244,    58479,     -2178,       1;
  1,  -6551,  325061,  -2160227, -2160227,    325061,   -6551,      1;
  1, -19672, 1736668, -19971304, 49441990, -19971304, 1736668, -19672, 1;

Mathematica

(* First program *)
q[x_, n_]= (1-x)^(n+1)*Sum[(2*m+1)^n*x^m, {m, 0, Infinity}];
t[n_, m_]:= t[n, m]= Table[CoefficientList[q[x, k], x], {k, 0, 15}][[n+1, m+1]];
p[x_, n_]:= p[x, n]= Sum[x^i*If[i==Floor[n/2] && Mod[n, 2]==0, 0, If[i <= Floor[n/2], (-1)^i*t[n, i], (-1)^(n-i+1)*t[n, i]]], {i, 0, n}]/(1-x);
Flatten[Table[CoefficientList[p[x, n], x], {n, 10}]]
(* Second Program *)
A060187[n_, k_]:= Sum[(-1)^(k-i)*Binomial[n, k-i]*(2*i-1)^(n-1), {i, k}];
T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, If[k<=Floor[n/2], T[n, k-1] +(-1)^k*A060187[n+2, k+1], T[n, n-k] ]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 18 2022 *)

Prog

(Sage)
def A060187(n, k): return sum( (-1)^(k-j)*(2*j-1)^(n-1)*binomial(n, k-j) for j in (1..k) )
@CachedFunction
def A225356(n, k):
    if (k==0 or k==n): return 1
    elif (k <= (n//2)): return A225356(n, k-1) + (-1)^k*A060187(n+2, k+1)
    else: return A225356(n, n-k)
flatten([[A225356(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 18 2022

Crossrefs

Cf A007318, A060187, A159041.

Sequence in context: A040486 A040485 A040484 * A291072 A174599 A291074

Adjacent sequences: A225353 A225354 A225355 * A225357 A225358 A225359

Keyword

sign,tabl

Author

Roger L. Bagula, May 07 2013

Extensions

Edited by N. J. A. Sloane, May 11 2013

Edited by G. C. Greubel, Mar 18 2022

Status

approved